Alignments of Random Points - An Estimate of The Probability of Alignments Existing By Chance

An Estimate of The Probability of Alignments Existing By Chance

Statistically, finding alignments on a landscape gets progressively easier as the geographic area to be considered increases. One way of understanding this phenomenon is to see that the increase in the number of possible combinations of sets of points in that area overwhelms the decrease in the probability that any given set of points in that area line up.

The number of alignments found is very sensitive to the allowed width w, increasing approximately proportionately to wk-2, where k is the number of points in an alignment.

For those interested in the mathematics, the following is a very approximate order-of-magnitude estimate of the likelihood of alignments, assuming a plane covered with uniformly distributed "significant" points.

Consider a set of n points in a compact area with approximate diameter d and area approximately d². Consider a valid line to be one where every point is within distance w/2 of the line (that is, lies on a track of width w, where w << d).

Consider all the unordered sets of k points from the n points, of which there are:

What is the probability that any given set of points is collinear in this way? Let us very roughly consider the line between the "leftmost" and "rightmost" two points of the k selected points (for some arbitrary left/right axis: we can choose top and bottom for the exceptional vertical case). These two points are by definition on this line. For each of the remaining k-2 points, the probability that the point is "near enough" to the line is roughly w/d, which can be seen by considering the ratio of the area of the line tolerance zone (roughly wd) and the overall area (roughly d²).

So, the expected number of k-point alignments, by this definition, is very roughly:

For n >> k this is approximately:

Now assume that area is equal to, and say there is a density α of points such that .

Then we have the expected number of lines equal to:

and an area density of k-point lines of:

Gathering the terms in k we have an areal density of k-point lines of:

Thus, contrary to intuition, the number of k-point lines expected from random chance increases much more than linearly with the size of the area considered.